the curiosity of mathematics

It has to do with curiosity.  It has to do with people wondering what makes something do something.  And then to discover, if you try to get answers, that they are related to each other–that things that make the wind make the waves, that the motion of water is like the motion of air is like the motion of sand.  The fact that things have common features.  It turns out more and more universal.  What we are looking for is how everything works.  What makes everything work.
—  Richard Feynman
the signs as animated movies

Aries: “Princess Mononoke”- passion, courage, risky, hard-hitting, exploration 

Taurus: “Ratatouille”- tasteful, dependability, ambition, down-to-earth, persistence   

Gemini: “Inside Out”- witty, enthusiastic, multi-dimensional, versatility, unpredictability 

Cancer: “Lilo & Stitch”- family, protectiveness, creativity, spontaneous, emotional

Leo: “The Lion King”- honor, pride, loyalty, friendship, warmth

Virgo: “Atlantis: The Lost Empire”- intelligence, selflessness, detailed, hard-working, moral

Libra: “Beauty and the Beast”- freedom, kindness, acceptance, sacrifice, innocence

Scorpio: “Coraline”- mystery, bravery, intuition, determination, secretive

Sagittarius: “Up”- adventure, fearless, big-hearted, devotion, acceptance 

Capricorn: “Ghost in the Shell”- caution, wisdom, mathematical, disciplined, contemplative

Aquarius: “Spirited Away”- unique, curiosity, honesty, justness, independence

Pisces: “Howl’s Moving Castle”- escapism, kindness, imagination, loving, compassion

Mathematical Curiosities

Kaprekar numbers 

Consider the number 7,777² = 60,481,729. If we separate this number into two parts, each four digits long, and then add them, we get 6,048 + 1,729 = 7,777, which is the same as the base we began with. 

The numbers need not be repetitive digits to be interesting. Consider the number 297:  297² = 88,209, and 88 + 209 = 297. 

Such a number is called a Kaprekar number, named after the Indian mathematician Dattaraya Ramchandra Kaprekar (1905-1986), who discovered such numbers. Some other Kaprekar numbers are 1; 9; 45; 55; 99; 703; 999; 2,223; 2,728; 4,879; 4,950; 5,050; 5,292; 7,272; 7,777; 9,999; 17,344; 22,222; 38,962; 77,778; 99,999; … ; 538,461; 857,143; …

We also have such things such Kaprekar triplets, which behave as follows: 

45³ = 91,125 with 9 + 11 + 25 = 45

Other Kaprekar triplets are 1; 8; 10; 297; 2,322.

While we are dealing with discoveries made by Kaprekar, we can admire the Kaprekar constant, 6,174. This constant arises when one takes any four-digit number and forms the largest and the smallest number from these digits, and then subtracts these two new numbers. Continuing this process will always eventually result in the number 6,174. When the number 6,174 is arrived at, we continue the process of creating the largest and the smallest number, and then taking their difference (7,614 - 1,467 = 6,174), we notice that we get back to 6,174. This is, therefore, called the Kaprekar constant. To demonstrate this with an example, we will consider the number 2,303:

  • The largest number formed with these digits is:            3,320.
  • The smallest number formed with these digits is:          0,233.
  • The difference is:       3,087.
  • The largest number formed with these digits is:            8,730.
  • The smallest number formed with these digits is:          0,378.
  • The difference is:       8,352.
  • The largest number formed with these digits is:            8,532.
  • The smallest number formed with these digits is:          2,358.
  • The difference is:       6,174.
  • The largest number formed with these digits is:            7,641.
  • The smallest number formed with these digits is:         1,467.
  • The difference is:       6,174.


Content credit: Mathematical Curiosities A Treasure Trove of Unexpected Entertainment

Minor correction: the Kaprekar constant only works for 4 digit numbers that have at least 2 different digits. (Rep-digits not allowed).Also, it was cool to see that there is a 3-digit Kaprekar constant: 495. And no other length of digit string has a constant.I would love to see a numberphile episode on this. (Thanks to @agingginger)

Mathematical Curiosities

This is a new series that I’m starting on this blog. All content from this series is credited to the book Mathematical Curiosities A Treasure Trove of Unexpected Entertainment.

A number pattern :

19 = 1 · 9 + (1 + 9)

29 = 2 · 9 + (2 + 9)

39 = 3 · 9 + (3 + 9)

49 = 4 · 9 + (4 + 9)

59 = 5 · 9 + (5 + 9)

69 = 6 · 9 + (6 + 9)

79 = 7 · 9 + (7 + 9)

89 = 8 · 9 + (8 + 9)

99 = 9 · 9 + (9 + 9)

And another curious pattern is : 

                                                  3² + 4² = 5²

                                     10² + 11² + 12² = 13² + 14² 

                          21² + 22² + 23² +24² = 25² + 26² + 27²

               36² + 37² + 38² +39² + 40² = 41² + 42² + 43² + 44²

    55² + 56² + 57² +58² + 59² + 60² = 61² + 62² + 63² +  64² +  65²

This can lead us to : 

1 + 2 = 3

4 + 5 + 6 = 7 + 8

9 + 10 + 11 + 12 = 13 + 14 + 15

16 + 17 + 18 + 19 + 20 = 21 + 22 + 23 + 24

and so on. 

One of my favorite professors once remarked that he can’t understand why people don’t treat mathematics lectures like sports games. Can you imagine if the professor walked in a said, “Today we’re going to introduce the idea of homology…” and the whole room started cheering? Someone yelled, “Thank god, FINALLY!” and someone else started crying tears of joy? 

I think about this every time a proof concludes in a really satisfying way. I look around at my classmates for a moment, half expecting them to start applauding. 

If I can do one thing with my teaching career in this lifetime, it is this: I want to incite riots on a regular basis in my classroom. You end proofs with QED. I hope to end them with “And that’s right, we see what’s going on now, Munkres didn’t prove that Lemma for nothing - It’s all set up, now, yes, you all see what he’s going to do, throw the lemma at it and the whole things just falls right into place CAN YOU BELIEVE THIS GUY?!” and I draw a square with a flourish, out of breath, hair sticking out at all angles, while applause explodes across the room. 

“I hope nobody had too big of a breakfast today, because….we’re introducing vector spaces.” 

*Hushed whispers*

“I know, I’m still in shock too, but today’s the day. I mean it! Vector spaces! Say it with me now, vector spaces. Vector spaces. VectoR SPACES. VECTOR SPACES! VECTOR SPACES!!!”

I want to be reported by surrounding classrooms at least once per semester. I want students from other classes to come join mine out of curiosity on the regular. I want mathematics to start fires in the minds of everyone around me, and I wouldn’t mind if we accidentally stared an actual fire… maybe, a small one, just once. 

The Zodiac Styles of Intelligence

♈ Aries: Bodily-Kinesthetic Intelligence (“Body Smart”)

Bodily kinesthetic intelligence is the capacity to manipulate objects and use a variety of physical skills. This intelligence also involves a sense of timing and the perfection of skills through mind–body union. Athletes, dancers, surgeons, and craftspeople exhibit well-developed bodily kinesthetic intelligence.
♉ Taurus: . Bodily-Kinesthetic Intelligence (“Body Smart”)

Bodily kinesthetic intelligence is the capacity to manipulate objects and use a variety of physical skills. This intelligence also involves a sense of timing and the perfection of skills through mind–body union
♊ Gemini: Linguistic Intelligence (“Word Smart”)
Linguistic intelligence is the ability to think in words and to use language to express and appreciate complex meanings. Linguistic intelligence is the most widely shared human competence and is evident in poets, novelists, journalists, and effective public speakers

Keep reading

A black hole is a mathematically defined region of spacetime exhibiting such a strong gravitational pull that no particle or electromagnetic radiation can escape from it. The theory of general relativity predicts that a sufficiently compact mass can deform spacetime to form a black hole. The boundary of the region from which no escape is possible is called the event horizon. Although crossing the event horizon has enormous effect on the fate of the object crossing it, it appears to have no locally detectable features. In many ways a black hole acts like an ideal black body, as it reflects no light. Quantum field theory in curved spacetime predicts that event horizons emit Hawking radiation, with the same spectrum as a black body of a temperature inversely proportional to its mass. This temperature is on the order of billionths of a Kelvin for black holes of stellar mass, making it essentially impossible to observe. Objects whose gravitational fields are too strong for light to escape were first considered in the 18th century by John Michell and Pierre-Simon Laplace. The first modern solution of general relativity that would characterize a black hole was found by German physicist and astronomer Karl Schwarzschild in 1916, although its interpretation as a region of space from which nothing can escape was first published by David Finkelstein in 1958. Long considered a mathematical curiosity, it was during the 1960s that theoretical work showed black holes were a generic prediction of general relativity. The discovery of neutron stars sparked interest in gravitationally collapsed compact objects as a possible astrophysical reality. 

Black holes of stellar mass are expected to form when very massive stars collapse at the end of their life cycle. After a black hole has formed, it can continue to grow by absorbing mass from its surroundings. By absorbing other stars and merging with other black holes, supermassive black holes of millions of solar masses (M☉) may form. There is general consensus that supermassive black holes exist in the centers of most galaxies. Despite its invisible interior, the presence of a black hole can be inferred through its interaction with other matter and with electromagnetic radiation such as visible light. Matter falling onto a black hole can form an accretion disk heated by friction, forming some of the brightest objects in the universe. If there are other stars orbiting a black hole, their orbit can be used to determine its mass and location. Such observations can be used to exclude possible alternatives (such as neutron stars). In this way, astronomers have identified numerous stellar black hole candidates in binary systems, and established that the core of the Milky Way contains a supermassive black hole of about 4.3 million M☉.

Transcendental numbers are not the roots of any algebraic equation. The existence of transcendental numbers was proven by Joseph Liouville in 1844.

In 1873 Charles Hermite proved that “e” is transcendental.

The German mathematician Ferdinand von Lindemann, in 1882, succeed in proving that π is transcendental. The area of a circle is π*r^2 (r = radius), that of a square is s^2 (s = side); consequently, the side of a square whose ares is equal to that of a circle with radius 1 is square root of π. A construction with straightedge and compass alone can give only lengths that are algebraic numbers, so Lindemann’s proof that π is transcendental was conclusive evidence that the age-old problem of squaring the circle is insolvable.

The transcendentality of e^π was proved in 1929. Yet, even if we know today that the number of transcendental numbers is infinite, there are still many irrational numbers, e.g. π^π, that defy our curiosity whether they are algebraic or transcendental.

Mathematical Curiosities

                           A few curious number patterns

One often asks, why do such unusual number patterns exist? Sometimes the best answer is that it is a peculiarity of the base-ten system. Although this may not satisfy everyone, for many it may suffice. Therefore, just enjoy and marvel at these patterns. 

A simple relationship:

                                                                1 = 1                                   = 1 · 1 = 1² 

                                                        1+2+1 = 2+2                               = 2 · 2 = 2²

                                                1+2+3+2+1 = 3+3+3                          = 3 · 3 = 3²

                                        1+2+3+4+3+2+1 = 4+4+4+4                      = 4 · 4 = 4²

                                1+2+3+4+5+4+3+2+1 = 5+5+5+5+5                  = 5 · 5 = 5²

                        1+2+3+4+5+6+5+4+3+2+1 = 6+6+6+6+6+6              = 6 · 6 = 6²

                1+2+3+4+5+6+7+6+5+4+3+2+1 = 7+7+7+7+7+7+7          =7 · 7 = 7²

        1+2+3+4+5+6+7+8+7+6+5+4+3+2+1 = 8+8+8+8+8+8+8+8      = 8 · 8 = 8²

1+2+3+4+5+6+7+8+9+8+7+6+5+4+3+2+1 = 9+9+9+9+9+9+9+9+9  = 9 · 9 = 9²


                                                         1 · 1 =  1

                                                     11 · 11 = 121

                                                 111 · 111 = 12321

                                             1111 · 1111 = 1234321            

                                         11111 · 11111 = 123454321

                                     111111 · 111111 = 12345654321

                                 1111111 · 1111111 = 1234567654321
                             11111111 · 11111111 = 123456787654321

                         111111111 · 111111111 = 12345678987654321


Another pattern appears when we replace the 1s with the 9s to get the following:

                                                     9 · 9 =  81

                                                 99 · 99 =  9801

                                             999 · 999 =  998001

                                         9999 · 9999 =  99980001

                                     99999 · 99999 =  9999800001

                                 999999 · 999999 =  999998000001

                             9999999 · 9999999 =  99999980000001

                         99999999 · 99999999 =  9999999800000001

                     999999999 · 999999999 =  999999998000000001


Content credit: Mathematical Curiosities A Treasure Trove of Unexpected Entertainment

Mathematical Curiosities

                                   More number patterns

  With 9s we can produce another aesthetically pleasing pattern shown below. 

                                         999,999 · 1 = 0,999,999

                                         999,999 · 2 = 1,999,998

                                         999,999 · 3 = 2,999,997

                                         999,999 · 4 = 3,999,996

                                         999,999 · 5 = 4,999,995   

                                         999,999 · 6 = 5,999,994

                                         999,999 · 7 = 6,999,993

                                         999,999 · 8 = 7,999,992

                                         999,999 · 9 = 8,999,991

                                         999,999 · 10 = 9,999,990


Here is another number pattern, where the number 9 is multiplied by a number representing the consecutive natural numbers- increasing by 1 each time- and then added to the initial natural numbers consecutively. 

                                                   0 · 9 + 1 = 1

                                                   1 · 9 + 2 = 11

                                                 12 · 9 + 3 = 111

                                               123 · 9 + 4 = 1111

                                             1234 · 9 + 5 = 11111

                                           12345 · 9 + 6 = 111111

                                         123456 · 9 + 7 = 1111111

                                       1234567 · 9 + 8 = 11111111

                                     12345678 · 9 + 9 = 111111111


We can consider number patterns that can be generated in a similar fashion, yet somewhat in the reverse of the previous one. However, this time the generated number consist of all 8s. 

                                                  0 · 9 + 8 = 8

                                                   9 · 9 + 7 = 88

                                                 98 · 9 + 6 = 888

                                               987 · 9 + 5 = 8888

                                             9876 · 9 + 4 = 88888

                                           98765 · 9 + 3 = 888888

                                         987654 · 9 + 2 = 8888888

                                       9876543 · 9 + 1 = 88888888

                                     98765432 · 9 + 0 = 888888888


Now that we have introduced the 8s in a rather dramatic fashion, we shall use them as the multiplier with the numbers consisting of increasing natural numbers, and each time adding successive natural numbers. Appreciating the number pattern shown here is more pleasing than trying to explain this phenomenon, which could conceivably detract from its beauty. 

                                                  1 · 8 + 1 = 9

                                                12 · 8 + 2 = 98

                                              123 · 8 + 3 = 987

                                            1234 · 8 + 4 = 9876

                                          12345 · 8 + 5 = 98765

                                        123456 · 8 + 6 = 987654

                                      1234567 · 8 + 7 = 9876543

                                    12345678 · 8 + 8 = 98765432

                                  123456789 · 8 + 9 = 987654321


Content credit: Mathematical Curiosities A Treasure Trove of Unexpected Entertainment